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MINISTRY OF EDUCATION AND TRAINING MINISTRY OF NATIONAL DEFENCE ACADEMY OF MILITARY SCIENCE AND TECHNOLOGY **************** TRAN DINH THONG IMPROVING THE CALCULATION SPEED OF TRANSMITH POWER MINIMUM PROBLEM FOR MULTI-ANTENNA WIRELESS TRANSMISSION NETWORK Specialization: Electronic Engineering Code No: 52 02 03 SUMMARY OF TECHNICAL DOCTORAL THESIS Ha Noi- 2020 This thesis has been completed at: ACADEMY OF MILITARY SCIENCE AND TECHNOLOGY Scientific supervisors: Dr Du Dinh Vien Dr Le Thanh Hai Reviewer 1: Prof Dr Bach Gia Duong University of Engineering and Technology, VNU Reviewer 2: Assoc Prof Dr Le Nhat Thang Posts and Telecommunications Institute of Technology Reviewer 3: Dr Vu Le Ha Academy of Military Science and Technology This thesis will be defended in front of Doctor Examining Committee held at Academy of Military Science and Technology at , date month year 2020 The thesis may be found at: - Library of Academy of Military Science and Technology - Vietnam National Library INTRODUCTION The urgency of thesis The diversification of new generation wireless transmission services in the context of limited spectrum resources is one of the concerns of the world's scientific community Conventional approaches focus on improving the efficiency of wireless transmission network There are three solutions to improve wireless network performance: Increasing the density of deployment of access points; adding more bands; improving the efficiency of using the spectrum Deploying additional access points as well as allocating new bands is costly and difficultly to implement Therefore, maximizing spectral performance on a given frequency band is an effective and feasible solution The optimization problem of the total antenna transmit power at the base station or relay node is one of the technical solutions to improve the efficiency of using the spectrum Hence, issues of minimum wireless transmission energy while guarantee a reasonable quality of service (QoS) based on application and development of modern optimization techniques which becoming increasingly urgent in the context of development media service rapidly The results from studies related to the minimum transmit power for multi-antenna wireless transmission networks will help scientists and policymakers to have effective technical solutions on designing of wireless transmission network The obtained results can be applied in new generation wireless transmission networks Therefore, the topic " Improving the calculation speed of transmit power minimum problem for multi-antenna wireless transmission network " was selected The objective Proposed solutions to improve the calculation speed for total transmit power minimum problem in multi-antenna wireless transmission network by using optimization techniques The subject and scopes Reseach on minimum problem of total transmit power with non-convex objective function which has SINR constraint in destination users for multiantenna wireless transmission model including base station broadcasting multipoint and multi-antenna relay wireless transmission relay network Suppose that power at the base station or relay node only takes into power of processing signals from the vector, the optimal weight matrix for beamforming at each antenna The signals from the sources originate in the same base frequency band with additive white Gaussian noise Research methodology Performing theoretical research, surveying research results in domestic and abroad, building objective functions with constraint conditions, applying mathematical transformations, simulating, analyzing and evaluating the results Mathematical theory and tools are mainly used in the thesis: Convex function theory, linear algebra theory related to matrix processing and analysis, optimization techniques, simulation software Matlab in combination with the support tools such as Sedumi, Yalmip, SDPT3 Scientific and practical significance of the thesis - The thesis focuses on the application of modern optimization techniques to solve the minimum problems of total transmit power with SINR boundary condition of destination users The proposals in the thesis help to find out the optimal value and improve the optimal speed at the same time Reseach on solutions combining techniques of space-time processing allows to improve performance, minimize interference effects as well as improve and overcome disadvantages - With the obtained results, the thesis contributes application and development of some optimal techniques in improving the performance of multi-antenna wireless transmission network Especially, applications in sensor network and new generation mobile communication network The thesis can be used as a useful reference in research, teaching and speciality training The results of the thesis are scientific basis for application in designing and planning problems for next generation wireless network Structure of the thesis The outline of thesis is composed of the beginning, chapters, the conclusion is as follows: Chapter 1: Minimum problem of total transmit power in multi-antenna wireless transmission network; Chapter 2: Improvement the calculation speed for the transmit power minimum problem of base stations transmitting multi-point broadcasting; Chapter 3: Improvement the calculation speed for the total transmit power minimum problem in multi-antenna relay wireless transmission network; Finally, the conclusion, evaluation and raising of issues need further study CHAPTER1 MINIMUM PROBLEM OF TOTAL TRANSMIT POWER IN MULTI-ANTENNA WIRELESS TRANSMISSION NETWORK 1.1 Overview The problem of minimum transmit power in wireless transmission network based on beamforming technique to improve using spectrum effectively This is a technique commonly used in wireless transmission networks combined with smart antenna technology for reducing access point consumption energy while maintaining quality of service (QoS) for the terminals In recent years, Massive MIMO is the one of the outstanding technical innovations in wireless transmission network system This technique opens up a new direction to improve the data transmission speed as well as the quality of the transmission Based on the survey of domestic and foreign studies, formulating transmit power optimization problems with constrain conditions in considering the problem: the scope of application of the model, factors affecting problem complex, using algorithm, survey model, data processing method are systematized as Figure 1.3 In the case of using mathematical transformations and a suitable search algorithm, the penalty function technique is quite a feasible and effective solution when applied Minimum transmit power problem Application range Factorial Mobile network Number of antennas SDP Concentrated relay Simulation Sensor network Number of users SDR Disconcentrated relay Experiment SONAR network SINR Random Vitual disconcentrat ed relay Data analysis Power Exact penalty function Co-operate relay Real network Spectral Base station Technique Model Method Figure 1.3: Systematize the problem of minimum transmit power for multi-antenna wireless transmission network 1.2 Existing issues and orientated research In the context of high-speed wireless transmission, the problems of minimizing transmit power for multi-antenna wireless transmission models with increasing number of antennas at base stations or relay nodes which increasing the complexity of the problem The simulation of highly complex problems in the current speed of computer processing takes a lot of time, affecting to design and plan of wireless transmission networks However, previous proposals that applied optimal techniques for the problem of minimum transmit power for multi-antenna wireless transmission network had two problems: Firstly, the Nonsmooth technique combined with the penalty function using the iterative algorithm for the minimum transmit power problems at the base station is very effective when compared with the random optimization techniques, SDR techniques However, previous proposals when conducting simulations only selected penalty factors randomly and increased penalty factors (according to exponential or linear function) after each iteration which leading to the determine of optimal value with convergence speed slowly It is impossible to determine the optimal value because the random selection process can not reach the optimal area in some cases Secondly, the problem of optimizing the total transmit power for multiantenna wireless relay transmision model has the increasing problem’s complexity as the number of increasing transceiver antennas In addition, the optimal variable is a matrix which has complexity is much greater than the minimum power penalty problem at the base station If using SDR technique through matrix vectorization, the problem complexity will increase and make it difficult to determine the optimal transmit power or convergence slowly 1.3 Mathematical theory 1.3.1 Convex function Currently, optimization techniques have been applied to the problem of objective function with boundary conditions Optimization theory has been thrived when S.Boyd and L Vanandberbere proposed the convex optimization theory The convex function has a very important that the local extreme is the global extreme Definition 1.1 A set C  N is a convex set if x, y  C satisfies x+(1- )y  C với    (1.1) Geometrically, a set C is called a convex set if the set of all points on the line connecting the two points x and y belongs to set C Figure 1.4: Convex and non-convex set N Definition 1.2 The function f:  is a convex function if (1.2) for every x, y  C and every  [0;1] Figure 1.13: Characteristic of convex 1.3.2 Penalty function method In the field of combinatorial optimization, the Lagrangian relaxation method (Avriel 1976, Fisher 1981, Reeves 1993) is implemented the reduction of the constraint conditions of the optimal problem using the cost function to advance to the convergence region In general, the optimal problem using the penalty function is expressed as (1.3): f (x) (1.3) satisfying the following conditions: x  A ; x  B where x is a variable vector to be searched, the constraint conditions x  A are easy to satisfy and constraint conditions x  B are relatively hard to satisfy Then the problem (1.3) is transformed: f (x)  p(d ( x, B)) (1.4) satisfying the following conditions: x A where, d (x, B) is data function that describes the distance from the vector value x to the region B and p (.) is a monotonically fine function that satisfy the condition p (0) = In fact, constraint conditions x  B is usually an equation or an inequality: gi (x)  0, i = 1, ,q (1.5) hi (x)  0, i = q + 1, ,m where q is the number of inequality conditions and m - q is the number of equality conditions 1.4 Constructing minimum power generation problems for diversity models 1.4.1 Spatial diversity model MIMO Figure 1.6: MIMO spatial diversity model MIMO spatial diversity model with gain channel matrix as shown in Figure 1.6 is shown by the equation: y  Hs  n , where H is the channel matrix M  M with hij showing the gain from the ith antenna to the jth on the receiver is a zero-weight Gaug complex number Where s is the signal vector of the source, y is signal vector at the destination, and n is the interference vector Then: h1,M   s1   n1   y1   h11       (1.6)          yM   hM ,1 hM ,M   sM   nM  1.4.2 Receiver antenna diversity model Figure 1.5 shows the technique of creating beamforming with M users on the source transmit independent signals from different locations in the transmitter space to the base station or relay node with M antennas The output signal from the kth beamforming is determined: yk  wkH x (1.7) where: x   x1 , , xM   C M 1 is a complex vector of the survey antenna array, T T wk   w k ,1 , , w k ,M   C M 1 is a complex weight vector for kth user Then, signal vector is received from the users on the source: x  hk sk  K  hs l 1,l  k l l n (1.8) where, hk and hl are the corresponding channel vectors from the source users to the receiving antenna arrays and n is the noise The optimal weight vector that can be obtained from maximizing the receiver SINR for each user is shown by the formula: SINRk  with Rs E (hk sk )(hk sk ) H  and trace( wkH Rs wk ) trace( wkH Ri  n wk ) Ri n (1.9) E (hl sl  v )(hl sl  n) H  are a covariance matrix of signals and interference noise Figure 1.7: Receiver antenna diversity model 1.3.3 Transmitter antenna diversity model Figure 1.8: Transmitter antenna diversity model In the case of transmitter antenna diversity as shown in Figure 1.8, if the signal x is transmitted from the antenna array at the base station then the receiver’s signal at the kth user is determined: rk  hkT x  v k (1.10) where, hk is the Gausian distribution channel between the base station and the kth receiver’s user and v k is additive white gaussian noise The weight vector wk  C M 1 is designed to minimize the transmit power at the base station, provided that the SINR constraint at the receiver's user is greater than the given fixed threshold The minimum problem of the total power generated at the base station is stated: M  trace(wk wkH ) wk (1.11) k 1 subject to:  trace(wkH Ri  n wk )  trace( wkH Rs wk )  , k = 1,2 , M (1.12) 1.5 Problems of minimum transmit power in wireless transmission network 1.5.1 Transmitted antenna diversity model in base station broardcasting 1.5.2 Transmission model has relay with AF processing method 1.6 Optimzation techniques 1.6.1 SDP optimzation technique 1.6.2 SDR optimzation technique 1.6.3 Nonsmooth combined with penalty function optimzation technique Considering the optimization problem with constraint condition minN  N f ( X ) (1.46)  X C subject to rank ( X )  (1.47) where X is a symmetric matrix PSD and f(X) is convex function The constrain condition (1.47) of the problem (1.40) is a non-convex function, the problem (1.46) is the optimal-NP-hard problem Condition (1.47) can be transformed into (1.48) trace( X )  max ( X )  where max ( X ) is the largest eigenvalue of matrix X If trace( X )  λ max ( X ) then trace( X )  max ( X ) , this means that only one unique eigenvalue of X satisfies: H (1.49) X  λmax ( X ) xmax xma x Where xmax is the unit eigenvalue vector ( x max  1) of X corresponding to the maximum eigenvalue max ( X ) Because max ( X ) is a convex function on the set of the Hermitian matrix, therefore, constrain condition trace( X )  max ( X ) is concave function In order to constraint condition to satisfy (1.48), trace( X )  max ( X ) should be small enough Based on the theory of the penalty function, the problem (1.46) is put into the form of problem using the penalty coefficient: minNN f ( X )  μ[trace( X )  λmax ( X )] (1.50) 0 X  C in which the penalty value   should be determined as appropriate to achieve a condition trace( X )  max ( X ) small enough To solve the problem (1.50), it is necessary to optimize both components f ( X ) and trace( X )  max ( X ) Regarding the speed of convergence during the process of formulating algorithm, the choice of coefficients and the steplength of penalty factors after each iteration will affect the convergence rate 1.7 Computational complexity of the optimization problem In fact, the minimum power problems for wireless transmission models presented in section 1.4 are combined optimal problems belonging to non-deterministic polynomial-time hard (NP-hard) problem Problems with a small number of operations can be solved by directly search methods Problems with great complexity using previously proposed optimization techniques often cannot find out the optimal solution or the determined value has a large tolerence with the real optimal value The complexity of the problem may be the number of calculations (including the number of memory accesses, or write to memory), but it also can be the total program execution time or the capacity of memory that needs to be allocated to run the program (spatial complexity) The procesing time for computer in algorithm performing is not only depending on the algorithm but also computer’s configuration In order to evaluate the effectiveness of an algorithm, considering the number of calculations that must be processed by algorithm running The effectiveness of an optimal technique is considered the optimal algorithm of construction techniques in many aspects: Effective application on problems: convex / non-convex problem, constraint/ non-constraint The speed of convergence of the algorithm: calculation time/ number of iterations Algorithm's solution quality: The results are obtained through stopping criterium or local/global solutions of the cost function The problems of optimal transmit power in multi-antenna wireless transmission network belong to the NP-hard problem Therefore, in order to analyze the computational complexity of algorithms, this section we use the complexity analysis formula of SDP problem SDP optimization problem with constrain condition is stated follow minN trace(CX ) (1.47) xR subject to trace( Ai X )  bi , X  0, i = 1, , M (1.48) where C and Ai are symmetric matrices, then the complexity of the problem is determined: ( n log n log(n/ε )) , with ε is the parameter of the algorithm In order to estimate the complexity of optimization problem SDP, it is necessary to determine the array parameter n of matrices in the object function and constraint condition Number of calculations depend on the size of the problem, In the survey models of the thesis, the complexity of problem depend on: the number of transmit antenna at base station, number of antennas at the relay node, number of transmitters and receivers, SINR threshold 1.8 Conclusion of chapter Research, development and apply effective optimization techniques for solving minimum power transmission problems in multi-antenna wireless transmission network with great scientific and practical significance In particular, the penalty function technique is selected to determine the minimum transmit power while improving the convergence speed compared with random, SDR optimization H 1/2 When trace(X) - λmax (X) is small enough, then X  max  X  xmax xmax , ma x ( X ) x max satisfies the constraint condition SINR of the problem (2.7) Therefore, the target of the problem should be optimally implemented so that trace(X) − λmax (X) reaches the minimum value Under these conditions, using the penalty function technique combined with the objective function, then the problem (2.15) becomes (2.18) f ( X )  trace( X )   trace( X )  max  X   0 X C N  N subject to trace( X H H i ) (2.19)   i , i = 1, 2, , M  i2 H On the other hand, the component gradient is xmax xma x , by using mathematical properties: H (2.20) max (Y )  max ( X )  trace((Y  X ) H xmax xmax ), Y  Based on the repeatability property, if X (k) is the optimal value of the kth iteration of the problem (2.18) with the largest eigenvalue λmax(X(k)) corresponding to the eigenvector X(k) Then, set up the problem (2.21) at the k + iteration,: trace( X )   trace( X )  max ( X ( k ) )  trace(( X  X ( k ) ) H x ( k ) x ( k ) H )  (2.21) SINRi  0 X C N  N subject to trace( X H H i )   i , i = 1, 2, , M  i2 become an SDP problem as (2.23) minN  N trace( X )    trace( X )  trace( X H x ( k ) x ( k ) H )  SINRi  0 X C (2.22) (2.23) subject to trace( X H H i ) (2.24)   i , i = 1, 2, , M  i2 When applying the Nonsmooth non-convex optimization technique, the penalty function needs to be performed in two stages to solve the problem (2.21) Specifically, at the initialization stage, choose an appropriate set of values (  , X (0) ) Proceeding to determine the optimal value X (k 1) that satisfied the constraint condition and given value  through solving the problem (2.21) The optimal stage selects the optimal value X (k 1) and fixes the value of µ from the initialization stage, continues to solve the problem (2.21) for determining the Xopt that satisfied the constrain condition The key steps of algorithm for Nonsmooth technique combined penalty function approach are outlined in Algorithm SINRi  Algorithm -% Initialization stage: - Initial step: Initialize proper  and X (0) to satisfy (2.21), set k = - Step k: Solve (2.21) to obtain its optimal solution X (k 1) + if trace(X ( k 1) )  max ( X ) (rank-one solution found) then Reset X (0) : X ( k1) Terminate, and output  and X (0) + else if trace(X ( k 1) )  max ( X ) (no improved solution found, no rank-one result) then Reset  : 2 and return to the initial step else Reset k : k  and X ( k )  X ( k 1) for the next iteration end if % Optimization stage: Set k = Solve (2.21) to obtain its optimal solution X ( k1) if trace(X ( k 1) )  trace(X ( k ) ) then Terminate, and output X opt  X ( k 1) else Reset k : k  and X ( k )  X ( k 1) Continue to the next iteration end if Output the final solution for problem (2.21) Many previous publications when applying algorithm 1, the authors often choose µ empirically but they did not calculate specifically There are many cases where the optimal value is determined, or some cases where the optimal value cannot be determined because selections does not proceed to the convergence area There always exists a value so that the problem (2.19) will be finished after only a few iterations The value is determined by: 0  max λT ν (2.25) ν 1 However  is very difficult to be found as the dual Lagrangian formulation of (2.25) is also NP-hard Hence the idea of finding the optimal value after only one iteration is almost impossible Fortunately, if  is obtained by the Lagrangian relaxation of (2.25) it is still useful as this value will help reduce significantly the number of iterations in NSM method The Lagrangian relaxation of (2.25) is established as, M 0 = max  λi i i2 (2.26) ( I N   hi hiH )  0, i = 1, 2, , M (2.27) i 1 subject to The key steps of our newly-devised approach are outlined as follow: Step 1: Obtain X (0) from (2.21) and 0 from (2.26) Step 2: Solve (2.21) and update  with very small amount The solution is also updated X ( k 1)  X ( k ) Step 3: When a rank-one solution is found, stop updating µ X(k) is still updated and converges after very few iterations 2.4 Simulation algorithm performance 2.4.1 SDR optimization algorithm performance 2.4.2 Random optimization algorithm performance 2.4.3 NSM1 optimization algorithm performance 2.4.3 NSM2 optimization algorithm performance 2.5 Numerical Results • Simulation Simulation compares SDR, random optimization techniques with Nonsmooth techniques (NSM1 and NSM2) with two criterias: the minimum total transmit power and the number of average iterations The simulation parameters are shown in Table 2.1 Table 2.1: Simulation parameters o N Parameter Value The number of users M on the receiver 16, 24 The number of antenna N at base station The number of iterations for determining penalty 30 factor µ The number of iterations at the optimal stage 20 Number of iterations ITE at each SNR threshold 500 SNR threshold (dB) 2, 4, 6, 8, 10 Initialized power value Pw for random engineering 2000 Stopping criterium 1 for NSM1 optimization 10-6 technique Stopping criterium 2 for NSM2 optimization 10-6 technique 10 Penalty factor µ in the initialization stage for NSM1 0.5 optimization technique 11 The steplength of penalty coefficient after each loop changes according to the exponential rule for NSM1 µ(k+1) = 1,5µ(k) technique - Total transmit power: Observing the simulation results in Figure 2.9 and Figure 2.10 shows that the Nonsmooth technique combined with the penalty function results closer to the lower boundary of SDR and better than the random technique In particular, the results of the total minimum transmit power of NSM1 and NSM2 techniques are approximately value(the graphs of NSM1 and NSM2 overlap) In the case of Figure 2.9, when the SNR threshold is less than dB, the optimal techniques give approximately results, but when the SNR is greater than dB, the effectiveness of the Nonsmooth technique with the penalty function has been demonstrate advantages This is also evident in the results of the graph of Figure 2.10 when increasing the number of users (M = 24) Detail data is shown in Tables 2.2 and 2.3 - Number of average iterations: Figures 2.11 and 2.12 describes the evaluation results, comparing the number of average iterations of the NSM2 technique with NSM1 Specifically, in the case of M = 16 and N = when the threshold SNR ≤ dB, the graphs of NSM1 and NSM2 in Figure 2.11 show that the number of average iterations of the two cases is the same Observing the results in the case of Figure 2.12, the number of average iterations of the NSM2 technique remains stable around the value of times However, the NSM1 graph at the threshold SNR  dB, the number of average iterations tended to decrease because the number of average iterations in the simulation to get the average result was not enough (about a few hundred times) Simulation results not reflect the theory Figure 2.9: Comparison of total transmitted power in the case of M = 16, N = Figure 2.11: Comparison of average number of iterations between NSM1 and NSM2 techniques (M = 16, N = 8) Figure 2.9: Comparison of total transmitted power in the case M = 24, N = Figure 2.11: Comparison of average number of iterations between NSM1 and NSM2 techniques (M = 24, N = 8) Remark: Through the results in simulation 1, it is confirmed that the advantages of Nonsmooth combined with penalty function technique compared to the random optimization technique and move towards optimal value of SDR technology The proposed optimization technique NSM2 has a fast convergence compared to the NSM1 optimization technique However, the number of iterations ITE run for each SNR threshold is not large enough That is reason why transmit power and the number of average iterations does not reflect the advantages of the proposed optimization technique • Simulation In order to demonstrate the effectiveness of the proposed NSM2 optimization technique, we conducted a simulation where the number of antennas at the base station N = and the number of users in destination varied among the three cases: M = 16, M = 24, M = 32 In particular, increase the number of iterations for each SNR threshold is 1000 times The simulation parameters are shown in Table 2.4 Table 2.4: Simulation parameters o N Parameter Value The number of users M on the receiver 16, 24, 32 The number of antenna N at base station The number of iterations itex for determining 30 penalty factor µ The number of iterations at the optimal stage 20 Number of iterations ITE at each SINR threshold 100 SNR threshold (dB) 2, 4, 6, 8, 10 Initialized power value for random engineering 2000 Stopping criterium 1 for NSM1 optimization 10-6 technique Stopping criterium 2 for NSM2 optimization 10-6 technique 10 Penalty factor µ0 in the initialization stage for 0.5 NSM1 optimization technique 11 The steplength of the penalty coefficient after each loop changes according to the exponential µ(k+1) = 1,5 µ(k) rule for NSM1 technique - Total transmit power: Observing the results on the graph in Figure 2.13 and detailed data in Tables 2.5, 2.6, 2.7 shows that the minimum power value of the proposed NSM2 technique is better in all cases at each SNR threshold Based on the calculation of the power ratio in Table 2.8, it was shown that when increasing the number of users (M = 32), the transmit power ratio of PNMS2/PNMS1 fluctuated stably from 0.93 to 0.95 When M = 16 and M = 24 the PNMS2/PNMS1 ranges from 0.96 to 1.0 Thus, the total transmit power has had good results when using the optimal technique proposed NSM2 Figure 2.13: Comparison of total transmitted power between NSM1 and NSM2 techniques Figure 2.14: Comparison of average number of iterations between NSM1 and NSM2 techniques Table 2.8: Transmit power ratio of PNMS2 / PNMS1 SNR dB dB dB dB 10 dB (M,N) = (16,8) 1.00 0.98 0.98 0.98 0.97 (M,N) = (24,8) 0.97 0.97 0.97 0.96 0.96 (M,N) = (32,8) 0.93 0.94 0.94 0.94 0.95 Table 2.12: Number of average iteration ratio ITENSM2/ITENSM1 SNR (M,N) = (16,8) (M,N) = (24,8) (M,N) = (32,8) dB 0.26 0.30 0.27 dB 0.27 0.27 0.29 dB 0.23 0.22 0.24 dB 0.22 0.21 0.24 10 dB 0.21 0.20 0.21 - Number of average iterations: The graph in Figure 2.14 and detailed data in tables 2.9, 2.10, 2.11 also demonstrate that the proposed NSM2 optimization technique has decreased significantly when compared to the NSM1 optimization technique Specifically, from the graph in Figure 2.14, the number of average iterations of the proposed technique is reduced by 3.5 to times and reaches a steady value when changes SNR threshold Table 2.12 shows the number of average iterations of ITENSM2/ITENSM1 between NSM2 and NSM1 techniques These data have shown that the ratio of ITENSM2 / ITENSM1 varies from 0.21 to 0.30 for the cases In particular, in case of M = 32, the number of average iterations of the NSM2 optimization technique has decreased from 71% to 79% compared to the NSM1 technique Remark: Through simulation results have shown that proposed techniques is not only determining the optimal value but also requiring additional calculations In the case of small scale problem (M, N is small number), increasing the computational complexity in the case of optimal parameter µ does not improve the convergence speed too much However, in case of large scale problem (M, N is large number), the optimization of penalty parameter µ become to greatly improves the convergence speed in determining the minimum transmit power 2.6 Conclusion of chapter In chapter 2, the thesis achieved some results as follow: - Formulated simulation algorithms using software Matlab in combination with SDPT3 and Yalmip tools to evaluate the results of the proposed optimization techniques compared to SDR and random optimization techniques - The simulation results have proved that SDR technique gives better results than random techniques In addition, the application of Nonsmooth technique combined with penalty function has results is not only asymptotic to SDR technology but also better the convergence speed However, the random selection of penalty coefficients and the steplength of the coefficients will affect the speed of convergence - The proposed optimization technique NSM2 has performed the optimal calculation of the penalty coefficient is not only determining the optimal value but also improving the speed of convergence The effectiveness of the proposed solution is even more evident when the scale of the problem increases in the number of users on the destination as well as the number of antennas at the base station at each SNR threshold In particular, the number of average iterations of NSM2 technique has decreased from 70% to 80% compared to NSM1 techniques The results in chapter have been published in two scientific papers [5] and [6] CHƯƠNG IMPROVEMENT CALCULATION SPEED FOR TOTAL TRANSMIT POWER MINIMUM PROBLEM IN MULTIANTENA WIRELESS RELAY TRANSMISSION NETWORK 3.1 The transmit power minimum problem in multi-antenna wireless transmission network 3.2 Multi-antenna wireless relay transmission model with AF processing protocol 3.2.1 Amplifier and forward method AF 3.2.2 Mathematical basis for formulating the total transmit power minimum problem 3.2.3 Formulating minimum problem for the multi-antenna relay model Consider a scenario of MIMO relaying communication as shown in Figure 3.2, where M sources communicate in pairs to other M receiver destinations with the assist of an N -antenna MIMO relay Since the use of the MIMO relay is to increase the coverage of communications links, it is reasonable to assume that the direct links between sources and destinations cannot be established A dualhop MIMO relay operating in the amplify-and-forward protocol mode is considered Let s  [s1 ,s , ,s M ]T  M be the vector of signals sent by M sources, which are assumed to be zero mean and component-wise independent with variance  s2  E[ si ] The MIMO Rayleigh flat-fading channels are considered in the chapter Let hi  [h i1 ,h i , ,h iN ]T  N , i  1, 2, , M be the uplink channel vector between source i and the relay while l j  [l j1 ,l j , ,l jN ]T  N , i  1, 2, , M denotes the downlink channel vector between the relay and destination j The signal vector s  [s1 ,s , ,s M ]T  C M is sent by M sources which assumed to be zero mean independent and variance  s2  E[ si ] For hi  [h i1 ,h i , ,h iN ]T  C N as the uplink channel vector between the user ith at transmitter and relay node, l j  [l j1,l j , ,l jN ]T  C N is the downlink channel vector between jth the relay and user jth at destination h11 l’11 h21 S1 l’21 Relay hN1 l’N1 D1 l’1M h1M l’2M h2M hNM l’NM SM DM Figure 3.2: MU-MIMO wireless relay model with AF processing method The signal is processed from the sources to the destination as follows: + On the first hop, all sources simultaneously transmit the signals to the relay The received signal at the relay is given by (3.13) yup  Hs  nr where H is the uplink channel matrix that contains all the channel vectors as its columns and nr  [n r1 ,n r , ,n rN ]T  N , i  1, 2, , N represents the additive noise at the relay receivers, which is assumed to be white Gaussian noise with zero mean and variance  r2  E[ nrn ] , + The second stage, relay transmits signal after processing to the users in destination Then, X is the optimal weight matrix is multiplied by the receiver signal at the relay Therefore, the relay sends the following signals to the destinations: (3.14) yrelay  Xyup  XHs  Xnr Accordingly, the received signal vector at the destinations is (3.15) yd  LXHs  LXnr  nd where nd is additive white Gaussian noise at the destination with element variance  d2 and L is the downlink channel matrix Specifically, the received signal at destination i can be presented by M ydi  l Xhi si   liT Xh j s j  liT Xnr  ndi T i j i (3.16) The total transmit power at the relay is determined as follows:  PT ( X )  E yrelay   trace( ( HH s H   r2 I N ) X H X ) (3.17) Signal to interference and noise ratio SINR at ith user at destination: SINRi ( X )   s2trace(li liH Xhi hiH X H )  s2  trace(li liH Xhi hiH X H )   r2trace(li liH XX H )   d2 (3.18) j i with li  ( li )* is is the complex conjugate of the downlink channel information on user i In relay communications, the key objective of the relay is to assist the communications between sources and destinations obtaining the predetermined quality of service (QoS) while minimizing the consuming power at the relay The problem of minimizing the total beamforming power under the SINR constraints is mathematically posed as trace(( s2 HH H   r2 I N ) X H X ) (3.19) X C NxN subject to SINRi ( X )   i , i = 1, 2, , M (3.20) 3.2.3 Formulating SDR optimization problem Total transmit power PT ( X ) in (3.19) is rewritten as formula (3.21): PT ( X )  trace(vec H ( X )vec( X )[( s2 HH H   r2 I N )T  I N ]) So, the SINR at destination i can be rewritten as  s2trace(vec H ( X )vec( X )[(hi hiH )T  ( li liH )]) SINRi ( X )  trace(vec H ( X )vec( X ) A  (li liH ))   d2 with A  ( M s hh i j H i i (3.21) (3.22)   r2 I N )T By substituting the nonlinear term vec H ( X )vec( X )  C N N by the Hermitian semi-definite variable X with X  C N optimization (3.19) is stated: N PT ( X ) 2 X C N N , the SDR of the nonconvex (3.23) subject to SINRi ( X )   i , i  1, 2, , M (3.24) When trace(vec H ( X )vec( X ) A  (li liH ))   d2  , the contrain condition (3.24) is expressed: trace(vec H ( X )[(A   i (hi hiH )T )  ( li liH )]vec( X ))   d2 i (3.25) 3.2.4 Formulating Spectral optimization problem By introducing a matrix Y so that its diagonal entries Y(i;i) play the role of nonlinear terms Y (i,i)  trace(li liH Xhi hiH X H ) , the SINR constraints (3.24) are now convex as (3.26):  i [ s2  trace(li liH Xhi hiH X H )   r2trace(li liH XX H )   d2 ]   s2Y (i,i)  0, j i (3.26) i = 1, 2, , M Define a linear mapping: Γ ( X ) : C NxN  C M by Γ ( X )  [l1H Xh1 , l 2H Xh2 , , l MH XhM ]T (3.27) Then the constraint condition in problem (3.19) for X is equivalent to the condition of (3.28) for (X, Y) only if (X, Y) are satisfied: Γ ( X )  Y (3.28) Y  H 0  Γ (X ) rank (Y )  (3.29) Actually (3.28) and (3.29) are equivalent to Y (i,i )  trace(li liH Xhi hiH X H ) Then, the optimization problem (3.23) is equivalent to (3.30) MxM PT ( X ) NxN X C ,Y C subject to  i [ s2  trace(li liH Xhi hiH X H )   r2trace(li liH XX H )   d2 ]   s2Y (i,i)  0, j i (3.31) i = 1, 2, , M Γ ( X )  Y (3.32) Y  H 0 Γ ( X )   rank (Y )  (3.33) There is one value 0  where any value of µ satisfies the condition:   0 , problem will be end after only a few iterations using the exactly penalty function, then (3.31) is equivalent to: (3.34) PT ( X )  [trace(Y )  max (Y )] X,Y subject to  i [ s2  trace(li liH Xhi hiH X H )   r2trace(li liH XX H )   d2 ]   s2Y (i,i)  0, j i (3.35) i = 1, 2, , M Γ ( X )  Y (3.36) Y  H 0 Γ ( X )   where Y defined by (3.32) SPECTRAL OPTIMIZATION ALGORITHM - Initialization: Choose initial feasible solution ( X (0) , Y (0) ) of (3.34) kth iteration: Solve convex program (3.43) to obtain the optimal solution ( X ( k 1) ,Y ( k 1) ) Given the tolerance level , if PT ( X ( k ) )  PT ( X ( k 1) )   (trace(Y ( k )  Y ( k 1) ))  f ( X ( k ) ,Y ( k ) )  f ( X ( k 1) ,Y ( k 1) )   Stop and output the solution ( X ( k ) ,Y ( k ) ) , Otherwise go to the next iteration Given the optimal value 3.3 Proposed Spectral optimization approach Firstly, each nonconvex element in SINR constraints is replaced by one linear variable (3.45) y (i )  trace(li liH Xhi hiH X H ); i  1, 2, , M Hence, additional constraints are required such that the equality (3.44) is satisfied These constraints can be expressed by  y (i) liH Xhi  Yi   H H  0 h X l i  i  rank (Y i ) = 1, i = 1, 2, , M (3.46) (3.47) The problem (3.43) can be formulated by M X ,Y PT ( X )    f (Yi ,Yi ) (k ) (3.49) i 1 subject to:  i ( s2  trace(li liH Xhi hiH X H )   r2trace(li liH XX H )   d2 )   s2 y (i)  (3.50) j i Y i  0, rank (Y i ) = 1, i = 1, 2, , M 3.4 Simulation algorithm performance 3.4.1 SDR optimization algorithm performance 3.4.2 SPO1 optimization algorithm performance 3.4.3 SPO2 optimization algorithm performance 3.5 Numerical Results (3.51) Table 3.1: Simulation parameters o N Parameter The number of users M at the transmitter and receiver Number of receiver and transmitter antennas N at the relay Number of iterations for each SINR threshold Penalty coefficient at the initialization stage SINR threshold (dB) Value 16, 24, 32 1000 1.0 2, 4, 6, 8, 10 10 11 Signal power (dB) Interference power at relay (dB) Noise power at each receiver user (dB) Stopping criterium spo1 for SPO1 optimization technique Stopping criterium spo2 for SPO2 optimization technique The steplength of the penalty coefficient after each iteration follow the exponential law -20 10-3 10-3 µ(k+1) = 2µ(k) Table 3.5: Computational time ratio between SPO1 and SPO2 technique SINR TSPO1/TSDR TSPO2/TSDR TSPO2/TSPO1 dB 0.52 0.29 0.56 dB 0.55 0.22 0.40 dB 0.64 0.19 0.30 dB 0.72 0.17 0.24 10 dB 1.05 0.16 0.15 Figure 3.10: Comparison of average steps between SPO1 and SPO2 techniques Figure 3.8: Comparison of total relay transmit power between SDR, SPO1, SPO2 techniques (M, N) = (4, 5) Figure 3.9: Comparing the average calculation time(s) Table 3.6: Average iterations ratio between SPO1 and SPO2 technique (M, N) = (4, 5) dB ITESPO2/ITESPO1 0.54 dB 0.42 dB 0.35 dB 0.30 10 dB 0.21 - Total transmit power: The graph in Figure 3.8 is also proved that the proposed SPO2 technique has produced the optimal value approximately with the other two techniques - Average calculation time: Figure 3.9 shows that the proposed technique "SPO2" has the calculation time faster than the "SPO1" technique from to times Specifically, Table 3.5 shows that the rate of calculation time of SPO1 compared with SDR is from 0.52 to 1.05 times, while the ratio of calculation time between SPO2 and SDR techniques ranges from 0.19 to 0.29 times The ratio of calculation time of SPO2 technique to SPO1 is from 0.15 to 0.56 Thereby, the time for calculating comparisons between SPO2 and SPO1 techniques has decreased from 44% to 85% - Number of average iterations: Figure 3.10 shows the number of average iterations of SPO1 and SPO2 techniquefor in each SINR threshold It can be seen that the proposed SPO2 technique only needs the number of iterations less than and maintains no more than when SINR changing from dB to 10 dB As can be seen from Table 3.6, when the SINR threshold increases, the average number of iterations between SPO2 and SPO1 techniques decreases from 0.54 to 0.21 times That means the average number of iterations of the SPO2 optimization technique has decreased from 46% to 79% as the SINR increased 3.6 Conclusion of chapter In chapter 3, the thesis has achieved some main results as follows: Simulation algorithm performance using Matlab software in combination with SDPT3, Yalmip tools to compare the proposed optimization technique SPO2 to SDR and SPO1 optimization techniques Evaluating and comparing the Spectral full-range optimization technique with SDR and Nonsmooth techniques using penalty function for multi-antenna relay wireless transmission model with AF processing protocol The simulation results show that the proposed optimization technique is not only finds out the optimal value but also reduces the calculation time and the number of average iterations The convergence speed of the proposed NSM2 technique has been significantly improved compared to the NSM1 technique In particular, the calculation time of the proposed optimization technique has been reduced from 44% to 85%, while the average number of steps has decreased from 46% to 79% The results in chapter have been published in scientific papers [1], [2], [4] and [7] CONCLUDING REMARKS The results of the thesis From the research content already done, the thesis has achieved the following main results: - Minimum problem of total transmit power for transmitting antenna diversity model: The thesis has studied the use of penalty function and iterative algorithm to determine the total minimum transmit power for multi-antenna radio transmission models The obtained results show that, using the penalty function technique for the broadcast base station model without cross-channel interference factor for optimal calculation of the penalty factor, the NSM2 technique find out good value than NSM1 technique with SNR threshold levels In addition, for the model of relay transmission transmission when using the penalty function through the use of linear auxiliary variables in SINR constraint conditions by the proposed SPO2 technique, the results show the minimum power value forward to SDR optimization techniques -Improving calculation speed problem: Using penalty function techniques has brought efficiency in raising the speed of convergence for the problem Specifically, for the base station transmission model, the NSM2 technique performed an optimal calculation of the penalty factor µ instead of random selection, which significantly reduced the average number of iterations Especially when the number of users increases, the average number of iterations of the proposed NSM2 technique decreases by 3.5 to times and reaches a steady value in cases when the SNR threshold level changes For multi-antenna relay wireless transmission model with AF processing method, SPO2 optimization technique has resulted in the calculation time and average number of repetitions reduced compared to SPO1 technique for SINR threshold varies from dB to 10 dB In particular, the calculation time of the proposed SPO2 optimization technique has been reduced from 44% to 85%, while the average number of iterations decreased from 46% to 79% The main contribution of the thesis Based on the research results, the thesis has new contributions as follows: + Proposed solution to improve the calculation speed for total transmit power minimum problem at base stations multi-point broadcast using Nonsmooth techniques combined with optimal penalties factor instead of randomly selected + Proposed solution to improve speed of calculating the total transmit power for multi-antenna relay wireless transmission network based on the development of Spectral optimization techniques using linear auxiliary variable with nonconvex function which has constant SINR condition Future research directions To continue researching, developing the achieved results and expanding the scope of the study, the next research direction of the thesis is proposed as follows: - Studying the problem of power balance between the antennas at the base station and relay node in the condition of optimizing the total transmit power to ensure the SINR level of the destination users - Research to minimize the maximum power on antenna brakes at the relay node with color noise conditions; - Research to build realistic test-bed models to verify the theoretical analysis and Monte-Carlo simulation results LIST OF PUBLICATIONS [1] Tran Dinh Thong, Du Dinh Vien, Le Thanh Hai, 2016, SDR optimization technique for MIMO wireless relay network, Journal of Science and Technology- Hanoi University of Industry, No.37, (12/2016), ISSN: 18593585, pp.13-37 [2] Tran Dinh Thong, Du Dinh Vien, Le Thanh Hai, 2017, Proposed optimal technique to solve beamforming problem for MIMO wireless network, Journal of Science and Technology- Hanoi University of Industry, No.38A, (02/2017), ISSN: 1859-3585, pp.55-59 [3] Tran Dinh Thong, Du Dinh Vien, Le Thanh Hai, 2018, Reseach and build object function for power optimum in wireless relay model, Journal of Science and Technology- Hanoi University of Industry, No.48, (10/2018), ISSN: 1859-3585, pp.123-128 [4] Tran Dinh Thong, Du Dinh Vien, Le Thanh Hai, 2018, Application of SDP technique to solve power in wireless relay MIMO, Journal of Science and Technology- Hanoi University of Industry, No.48, (10/2018), ISSN: 1859-3585, pp.138-142 [5] Tran Dinh Thong, Du Dinh Vien, Le Thanh Hai, 2019, A newly developed optimization method for multicast transmission, The 4th International Conference on Reseacher in Intelligent Computing in Engineering (RICE2019) [6] Tran Dinh Thong, Du Dinh Vien, Le Thanh Hai, Phan Huy Anh, Tran Hoang Linh, 2019, Fast convergence Nonsmooth optimization approach fof beamforming problems in multi-antenna wireless transmission scenarios, Journal Military Science anh Technology, No.48, (10/2019), ISSN: 18591043, pp.95-101 [7] Tran Dinh Thong, Du Dinh Vien, Le Thanh Hai, Phan Huy Anh, 2019, Improve convergence speed for optimizaton of relay power problem using Spectral optimizaton technique in multi-antenna wireless transmision, Journal Military Science anh Technology, Special Issue, No.63A, (11/2019), ISSN: 1859-1043, pp.39-51 ... noise Figure 1.7: Receiver antenna diversity model 1.3.3 Transmitter antenna diversity model Figure 1.8: Transmitter antenna diversity model In the case of transmitter antenna diversity as shown... 2.12 shows the number of average iterations of ITENSM2/ITENSM1 between NSM2 and NSM1 techniques These data have shown that the ratio of ITENSM2 / ITENSM1 varies from 0.21 to 0.30 for the cases... PT ( X ) in (3.19) is rewritten as formula (3.21): PT ( X )  trace(vec H ( X )vec( X )[( s2 HH H   r2 I N )T  I N ]) So, the SINR at destination i can be rewritten as  s2trace(vec H ( X